Estimation of parameters [in probabilistic models] One way to estimate parameters in simple probabilistic models is to compute the mean and other moments of the data and then to work out what values of the parameters will reproduce these. More general is the maximum likelihood method in which one finds the values of the parameters which maximize the probability of generating the observed data from the model. … If one starts with a priori probability distributions for all parameters, then Bayes's Theorem on conditional probabilities allows one to avoid the arbitrariness of methods such as maximum likelihood and explicitly to work out from the observed data what the probability is for each possible choice of parameters in the model.

But in a sense one of the most surprising features of the facing page is that the fluctuations it shows are so violent. One might have thought that in going say from f[2000] to f[2001] there would only ever be a small change. … But in general one suspects that all these rules can be thought of as being like simple computer programs that take some representation of n as their input.

And indeed in any system, the amount of time over which the details of initial conditions can ever be considered the dominant source of randomness will inevitably be limited by the level of separation that exists between the large-scale features that one observes and small-scale features that one cannot readily control. … And what the pictures demonstrate is that even if the initial position of this mass is changed by just one part in a hundred million, then within 50 revolutions of the large masses the trajectory of the small mass will end up being almost completely different.

But it turns out that as soon as one tries to make a more complete model, there are immediately an immense number of issues that arise, and it is difficult to know which are really important and which are not. At a basic level, one knows that snowflakes are formed when water vapor in a cloud freezes into ice, and that the structure of a given snowflake is determined by the temperature and humidity of the environment in which it grows, and the length of time it spends there. … Various models of snowflake growth exist in the standard scientific literature, typically focusing on one or two of these effects.

But beyond this there is a surprisingly universal conviction that any significant property that one sees in any organism must be there because it in essence serves a purpose in maximizing the fitness of the organism. … And indeed as one looks at more and more complex features of biological organisms—notably texture and pigmentation patterns—it becomes increasingly difficult to find any credible purpose at all that would be served by the details of what one sees.

On the basis of traditional biological thinking one would tend to assume that whatever complexity one saw must in the end be carefully crafted to satisfy some elaborate set of constraints. … So how can one tell if this is really the case?

instructions one can get all sorts of complex behavior is similar to the phenomenon we have seen in cellular automata. … And one might hope that it would be possible to call on some existing kind of intuition to understand such a fundamental phenomenon. … And by absorbing these examples, one is in the end able to develop an intuition that makes the basic phenomena that I have discovered seem somehow almost obvious and inevitable.

And comparing the patterns in these pictures with patterns on actual mollusc shells, one notices the remarkable fact that the range of patterns that occur in the two cases is extremely similar. … Mollusc shells are almost unique in having patterns that are built up one line at a time; much more common is for patterns to develop all at once all over a surface. … But since different parts of an animal usually grow at different rates, the final pattern that one sees on an adult animal ends up being scaled differently in different places—so that, for example, the pattern is smaller in scale on the head of an animal, since the head grows more slowly.

But so far as one can tell the only types of rules that were tried were ones associated with standard geometry and arithmetic. … But the particular rules that were found to work were fairly sophisticated ones based on particular kinds of mathematical equations.

And instead what I expect is much like we have seen many times before in this book: that even though at the lowest level there is no direct correspondence between space and time, such a correspondence nevertheless emerges when one looks in the appropriate way at larger scales of the kind probed by practical experiments. … But one of the features of a cellular automaton is that it is set up to update all of its cells together, as if at each tick of some global clock. … At first it may seem bizarre, but one possibility that I believe is ultimately not too far from correct is that the universe might work not like a cellular automaton in which all cells get updated at once, but instead like a mobile automaton or Turing machine, in which just a single cell gets updated at each step.